2006
DOI: 10.1090/s0002-9939-06-08243-8
|View full text |Cite
|
Sign up to set email alerts
|

𝖫^{2}-summand vectors in Banach spaces

Abstract: Abstract. The aim of this paper is to study the set L 2 X of all L 2 -summand vectors of a real Banach space X. We provide a characterization of L 2 -summand vectors in smooth real Banach spaces and a general decomposition theorem which shows that every real Banach space can be decomposed as an L 2 -sum of a Hilbert space and a Banach space without nontrivial L 2 -summand vectors. As a consequence, we generalize some results and we obtain intrinsic characterizations of real Hilbert spaces. BackgroundWe say tha… Show more

Help me understand this report

Search citation statements

Order By: Relevance

Paper Sections

Select...
2
2
1

Citation Types

0
5
0

Year Published

2008
2008
2023
2023

Publication Types

Select...
5

Relationship

1
4

Authors

Journals

citations
Cited by 8 publications
(5 citation statements)
references
References 6 publications
0
5
0
Order By: Relevance
“…REMARK 2.2. In [1], it is proved that the set L 2 X of all L 2 -summand vectors of a real Banach space X is a closed vector subspace (in fact, it is a Hilbert subspace), that is, L 2 -complemented in X (that is, there exists a closed vector subspace M of X such that X = L 2 X ⊕ 2 M). In addition, it is shown that M = {ker(e * ) : e ∈ L 2 X }, where each e * is the L 2 -summand functional associated to each e. REMARK 2.3.…”
Section: Main Results and Consequencesmentioning
confidence: 99%
See 1 more Smart Citation
“…REMARK 2.2. In [1], it is proved that the set L 2 X of all L 2 -summand vectors of a real Banach space X is a closed vector subspace (in fact, it is a Hilbert subspace), that is, L 2 -complemented in X (that is, there exists a closed vector subspace M of X such that X = L 2 X ⊕ 2 M). In addition, it is shown that M = {ker(e * ) : e ∈ L 2 X }, where each e * is the L 2 -summand functional associated to each e. REMARK 2.3.…”
Section: Main Results and Consequencesmentioning
confidence: 99%
“…A vector e of a real Banach space X is said to be an L 2 -summand vector if there exists a closed vector subspace M of X such that X = ‫ޒ‬e ⊕ 2 M; in other words, λe + m 2 = λe 2 + m 2 for every λ ∈ ‫ޒ‬ and every m ∈ M. If e = 0, then the functional e * ∈ X * such that e * (e) = 1 and M = ker(e * ) is called the L 2 -summand functional associated to e. It satisfies e * = 1 e , where e * is an L 2 -summand vector of X * and X * = ‫ޒ‬e * ⊕ 2 ker ( e), where e denotes the element e in the bidual X * * (note that the L 2 -summand functional associated to e * is e.) We refer the reader to [1] and [2] for a wider perspective about L 2 -summand vectors.…”
Section: Introductionmentioning
confidence: 99%
“…It is easy to verify that u + 1 2 B X ∩ S X = [p, q], where p = − 1 2 , 1 and q = 1 2 , 1 . Let z be an arbitrary point in [p, q] and y be an arbitrary point in 1 2 B X that is isosceles orthogonal to z. On the one hand, z ⊥ I ( y , 0).…”
Section: The Sets Hmentioning
confidence: 99%
“…holds, then M is said to be an L 2 -summand subspace (cf. [1]). Note that, when M is an L 2 -summand subspace, N is uniquely determined.…”
Section: The Sets Hmentioning
confidence: 99%
See 1 more Smart Citation